1. Field of the Invention
This invention relates to the field of spacecraft vehicle control and, more specifically, to a method and apparatus for avoiding gyroscope array singularities.
2. Description of the Related Art
Control moment gyroscope (CMG) arrays are used for spacecraft attitude control due to their torque-producing capability. A typical attitude control system includes an array of three or more CMGs. Each CMG within the array includes a rotor spinning about an axis mounted to an actuated gimbal. The rotational motion of the rotor and gimbal creates an angular momentum vector and a torque vector. The torque vector is orthogonal to the angular momentum vector. The rate of rotation of the gimbal controls the torque vector, producing movement in a desired direction when the torque is not zero.
Controlling attitude for a spacecraft requires at least three degrees of freedom. Thus, a typical attitude adjustment system may use an array of at least three CMGs that together output a combined torque. A corrective torque vector can be calculated by comparing the desired attitude of the spacecraft with the actual spacecraft attitude to generate a corrective torque vector μ.
Corrective torque vector μ may then be used to calculate the gimbal rate {dot over (δ)} necessary to correct attitude. This calculation may be expressed as A{dot over (δ)}=μ, where A is a Jacobian matrix as a function of the gimbal angle values δ for the CMGs in the CMG array, {dot over (δ)} is a gimbal rate and μ is the corrective torque vector. Without performing null motion calculations, the gimbal rate {dot over (δ)} can be calculated by a pseudo-inverse steering law equation {dot over (δ)}=AT(AAT)−1μ where {dot over (δ)} is the gimbal rate, A is a Jacobian matrix determined by the gimbal angle values δ for each of the CMGs in the CMG array, AT is the Jacobian matrix A transposed, and μ is the corrective torque vector.
An undesirable condition, known as a singularity or a singular state, occurs when gimbals of a CMG array are respectively oriented so that each of the torque vectors produced by moving the gimbals are parallel to each other. The pseudo-inverse steering law equation is unusable when a CMG array is in a singular state. When a CMG array is in a singular state, the determinant D of Jacobian matrix A is zero. For a CMG array using three CMGs, a Jacobian matrix A has the form
      [                                        L            1                                                L            2                                                L            3                                                            M            1                                                M            2                                                M            3                                                            N            1                                                N            2                                                N            3                                ]    ,where Li denotes the ith column value of row L, Mi denotes the ith column value of row M and Ni denotes the ith column value of row N. Calculation of determinant D uses the equation D=L1(M2N3−M3N2)−L2(M1N3−M3N1)+L3(M1N2−M2N1).
Various calculations are known in the art for addressing singularities. These methods include singularity-robust inverse techniques, path planning methods, preferred gimbal angle methods and the use of variable speed CMGs (VSCMGs). However, these methods each exhibit various limitations. For example, the singularity-robust inverse technique of Roser et al. cannot not be used for all angle sets produced by a CMG array. (X. Roser, M. Sghedoni, “Control Moment Gyroscopes (CMG's) and their Application in Future Scientific Missions,” Spacecraft Guidance, Navigation and Control Systems, Proceedings of the 3rd ESA International Conference, 26-29 Nov. 1996, pp. 523-528, Noordwijk, the Netherlands.)
U.S. Pat. No. 6,039,290 to Wie et al. teaches a method using non-zero elements in the off-diagonal element of the positive definite diagonal matrix P. However, the choice of matrix P is arbitrary and does not guarantee quick transients of singular states.
U.S. Pat. No. 6,131,056 to Bailey et al. teaches a method of using an open-loop signal when the CMG array becomes singular. This method is computationally expensive and may result in a large control error due to disturbances.